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International Journal of Research p-I SSN: 2348-6848 e-I SSN: 2348-795X Available at Volume 04 Issue 03 https://edupediapublications.org/journals Ma rch 2017 A Study on Methods of Contour Integration Of Complex Analysis Abdulsattar Abdullah Hamad, Master of mathematics Department of mathematics College of Science, Acharya nagarjuna University Email id: [email protected] ABSTRACT: -Application of the Cauchy integral formula Complex analysis is considered as a -Application of the residue theorem. powerful tool in solving problems in One method can be used, or a combination mathematics, physics, and engineering. In of these methods or various limiting the mathematical field of complex analysis, processes, for the purpose of finding these contour integration is a method of evaluating integrals or sums. certain integrals along paths in the complex INTRODUCTION: plane. Contour integration is closely related Cauchy is considered as a principal founder to the calculus of residues, a method of of complex function theory. However, the complex analysis. One use for contour brilliant Swiss mathematician Euler (1707- integrals is the evaluation of integrals along 1783) also took an important role in the field the real line that are not readily found by of complex analysis. The symbol for √−1 using only real variable methods. Contour was first used by Euler, who also introduced integration methods include: 휋 as the ratio of the length of a circumference of a circle to its diameter and -Direct integration of a complex –valued 푒 as a base for the natural logarithms, function along a curve in the complex plane (a contour) respectively. He also developed one of the Available online: https://edupediapublications.org/journals/index.php/IJR/ P a g e | 622 International Journal of Research p-I SSN: 2348-6848 e-I SSN: 2348-795X Available at Volume 04 Issue 03 https://edupediapublications.org/journals Ma rch 2017 most useful formulas in mathematics. That laid the foundation from which another is Euler’s formula 푒푖휃 = cos 휃 + sin 휃 , mathematical giant, Cauchy, developed the which can be derived from De Moivre’s full scale theory of complex analysis and its theorem [1]. applications. Letting 휃 = 휋 in Euler’s formula, we obtain Complex analysis is essentially the study of the equation 푒푖휋 + 1 = 0. The equation, complex numbers, their derivatives, and 푒푖휋 + 1 = 0 , contains what some integrals with many other properties. On the mathematicians believe as the most other hand, number theory is concerned with significant numbers in all of mathematics, the study of the properties of the natural i.e. 푒, 휋, And the two integers o and 1. It numbers. These two fields of study seen to 휋 푖 − be unrelated to each other. However, in also gave the remarkable result = 푒 2 , number theory, it is sometimes impossible to which means that an imaginary power of an prove or solve some problems without the imaginary number can be a real number , use complex analysis. The same is true in and he showed that the system of complex many problems in applied mathematics, numbers is closed under the elementary physics and engineering. In a calculus transcendental operations. Moreover, Euler course, the exact value of some real –valued was the first person to have used complex definite integrals cannot be computed by analysis methods for trying to prove finding its anti-derivative because some Fermat’s last theorem, and he actually anti-derivative is impossible to find in terms proved the impossibility of integer solutions of elementary functions. In these cases, the of 푥 3 + 푦3 = 푧3. Euler’s contribution to integrals are only calculated numerically. complex analysis is significant. His work Available online: https://edupediapublications.org/journals/index.php/IJR/ P a g e | 623 International Journal of Research p-I SSN: 2348-6848 e-I SSN: 2348-795X Available at Volume 04 Issue 03 https://edupediapublications.org/journals Ma rch 2017 Amazingly, complex analysis enables us to The French mathematician Augustin-Louis evaluate many of these integrals. One Cauchy (1789-1857) brought great important application of complex analysis is contributions to mathematics, providing to find the values of real definite integrals by foundations for mathematical analysis, calculating contour integrals in the complex establishing the limit concept and general plane [2]. theory of convergence, and defining the definite integral as the limit of a Sum. A real definite integral is an integral 푏 푓(푥)푑푥 , where 푓(푥) is a real-valued Cauchy also devised the first systematic ∫푎 function with real numbers a and b. Here, a theory of complex numbers. Especially, his best-known work in complex function general definite integral is taken in the theory provides the Cauchy integral theorem complex plane, and we have 푓(푧)푑푧 ∫푐 as a powerful tool in analysis. Moreover, his where c is a contour and z is a complex discovery of the calculus of resides is variable. Contour integration provides a extremely valuable and its applications are method for evaluating integrals by marvelous because it can be applied to the investigating singularities of the function in evaluation of definite integrals, the domains of the complex plane. The theory of summation of series , solving ordinary and the functions of a real variable had been partial differential equations , difference and developed by Lagrange, but the theory of algebraic equations, to the theory of functions of a complex variable had been symmetric functions , and moreover to achieved by the efforts of Cauchy. mathematical physics. Available online: https://edupediapublications.org/journals/index.php/IJR/ P a g e | 624 International Journal of Research p-I SSN: 2348-6848 e-I SSN: 2348-795X Available at Volume 04 Issue 03 https://edupediapublications.org/journals Ma rch 2017 In the resume, Cauchy gave his definition of CONTOUR INTEGRAL the derivative as a limit and defined the We turn now to integrals of complex-valued definite integral as the limit of a sum. As a functions 푓 of the complex variable 푧. Such consequence, Weirstrass and Riemann an integral is defined in terms of the values extended his work on complex function 푓(푧) along a given contour 퐶 , extending theory, and also Riemann and Lebesgue from a point 푧 = 푧1 to a point 푧 = 푧2 in improved on his definition of the integral. In the complex plane. It is, therefore, a line his book, he also retained the mean-value integral; and its value depends, in general, theorem much as Lagrange had derived it. on the contour 퐶 as well as on the function Unlike Lagrange, Cauchy used tools from 푓. It is written integral calculus to obtain Taylor’s theorem 푧2 [3]. ∫ 푓(푧)푑푧 표푟 ∫ 푓(푧) 푑푧, 푐 푧1 Cauchy also did more with convergence The latter notation often being used when properties for Taylor’s series. In the ‘’ cours the value of the integral is independent of d’analyse ‘’, he had formulated the Cauchy the choice of the contour taken between two criterion independently of Bolzano early in fixed end points. While the integral may be 1817. In his writings, Cauchy proved and defined directly as the limit of a sum, we often utilized the ratio, root, and integral choose to define it in terms of a definite tests, thereby establishing the first general integral . Suppose that the equation theory of convergence. Among those great works, the importance of Cauchy’s residue 푧 = 푧(푡) (푎 ≤ 푡 ≤ 푏) (1.2.5.1) theory cannot be underestimated. Available online: https://edupediapublications.org/journals/index.php/IJR/ P a g e | 625 International Journal of Research p-I SSN: 2348-6848 e-I SSN: 2348-795X Available at Volume 04 Issue 03 https://edupediapublications.org/journals Ma rch 2017 Represents a contour 퐶, extending from a neighborhood of 푧0 . It follows that if 푓 is point 푧1 = 푧(푎) to a point 푧2 = 푧(푏). We analytic at a point 푧0, it must be analytic at assume that 푓[푧(푡)] is piecewise continuous each point in some neighborhood of 푧0 . A on the interval 푎 ≤ 푡 ≤ 푏 and refer to the function f is analytic in an open set if it has a function 푓(푧)as being piecewise continuous derivative everywhere in that set. If we on 퐶 [4]. We then define the line integral, or should speak of a function 푓 that is analytic contour integral, off along 퐶 in terms of the in a set 푆 which is not open, it is to be parameter 푡: understood that f is analytic in an open set containing 푆. 푏 ∫ 푓(푧) 푑푧 = ∫ 푓[푧(푡)] 푧′(푡)푑푡. (1.2.5.2) 푐 푎 Note that the function 푓(푧) = 1 is analytic at 푧 Note that since 퐶 is a contour, 푧′(푡) is also each nonzero point in the finite plane. But piecewise continuous on 푎 ≤ 푡 ≤ 푏; and the function 푓(푧) = | 푧|2 is not analytic at so the existence of integral (1.2.5.2) is any point since its derivativeexists only at ensured. The value of a contour integral is 푧 = 0 and not throughout any invariant under a change in the neighborhood. An entire function is a representation of its contour. function that is analytic at each point in the ANALYTIC FUNCTIONS entire finite plane. Since the derivative of a polynomial exists everywhere, it follows We are now ready to introduce the concept that every polynomial is an entire function. of an analytic function. A function 푓 of the If a function 푓 fails to be analytic at a point complex variable 푧 is analytic at a point 푧0 if 푧0 but is analytic at some point in every it has a derivative at each point in some neighborhood of 푧0 , then 푧0 is called a Available online: https://edupediapublications.org/journals/index.php/IJR/ P a g e | 626 International Journal of Research p-I SSN: 2348-6848 e-I SSN: 2348-795X Available at Volume 04 Issue 03 https://edupediapublications.org/journals Ma rch 2017 singular point, or singularity, of 푓.
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